DOCSLIB.ORG

Accelerated Gibbs Sampling of Normal Distributions Using Matrix Splittings and Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Accelerated Gibbs Sampling of Normal Distributions Using Matrix Splittings and Polynomials Bernoulli 23(4B), 2017, 3711–3743 DOI: 10.3150/16-BEJ863 Accelerated Gibbs sampling of normal distributions using matrix splittings and polynomials COLIN FOX1 and ALBERT PARKER2 1Department of Physics, University of Otago, Dunedin, New Zealand. E-mail: [email protected] 2Center for Biofilm Engineering, Department of Mathematical Sciences, Montana State University, Boze- man, MT, USA. E-mail: [email protected] Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss–Seidel iterative solution of linear equations in the precision matrix. Specifically, the iteration operators, the conditions under which convergence occurs, and geomet- ric convergence factors (and rates) are identical. These results hold for arbitrary matrix splittings from classical iterative methods in numerical linear algebra giving easy access to mature results in that field, including existing convergence results for antithetic-variable Gibbs sampling, REGS sampling, and gener- alizations. Hence, efficient deterministic stationary relaxation schemes lead to efficient generalizations of Gibbs sampling. The technique of polynomial acceleration that significantly improves the convergence rate of an iterative solver derived from a symmetric matrix splitting may be applied to accelerate the equivalent generalized Gibbs sampler. Identicality of error polynomials guarantees convergence of the inhomogeneous Markov chain, while equality of convergence factors ensures that the optimal solver leads to the optimal sampler. Numerical examples are presented, including a Chebyshev accelerated SSOR Gibbs sampler ap- plied to a stylized demonstration of low-level Bayesian image reconstruction in a large 3-dimensional linear inverse problem. Keywords: Bayesian inference; Gaussian Markov random field; Gibbs sampling; matrix splitting; multivariate normal distribution; non-stationary stochastic iteration; polynomial acceleration 1. Introduction The Metropolis–Hastings algorithm for MCMC was introduced to main-stream statistics around 1990 (Robert and Casella [48]), though prior to that the Gibbs sampler provided a coherent approach to investigating distributions with Markov random field structure (Turcinˇ [60], Grenan- der [32], Geman and Geman [25], Gelfand and Smith [23], Besag and Green [11], Sokal [58]). The Gibbs sampler may be thought of as a particular Metropolis–Hastings algorithm that uses the conditional distributions as proposal distributions, with acceptance probability always equal to1(Geyer[26]). In statistics, the Gibbs sampler is popular because of ease of implementation (see, e.g., Roberts and Sahu [51]), when conditional distributions are available in the sense that samples may be drawn from the full conditionals. However, the Gibbs sampler is not often presented as an ef- ficient algorithm, particularly for massive models. In this work, we show that generalized and 1350-7265 © 2017 ISI/BS 3712 C. Fox and A. Parker accelerated Gibbs samplers are contenders for the fastest sampling algorithms for normal target distributions, because they are equivalent to the fastest algorithms for solving systems of linear equations. Almost all current MCMC algorithms, including Gibbs samplers, simulate a fixed transition kernel that induces a homogeneous Markov chain that converges geometrically in distribution to the desired target distribution. In this aspect, modern variants of the Metropolis–Hastings algorithm are unchanged from the Metropolis algorithm as first implemented in the 1950s. The adaptive Metropolis algorithm of Haario et al. [34] (see also Roberts and Rosenthal [50]) is an exception, though it converges to a geometrically convergent Metropolis–Hastings algorithm that bounds convergence behaviour. We focus on the application of Gibbs sampling to drawing samples from a multivariate normal distribution with a given covariance or precision matrix. Our concern is to develop generalized Gibbs samplers with optimal geometric, or better than geometric, distributional convergence by drawing on ideas in numerical computation, particularly the mature field of computational linear algebra. We apply the matrix-splitting formalism to show that fixed-scan Gibbs sampling from a multivariate normal is equivalent in fundamental ways to the stationary linear iterative solvers applied to systems of equations in the precision matrix. Stationary iterative solvers are now considered to be very slow precisely because of their geometric rate of convergence, and are no longer used for large systems. However, they remain a basic building block in the most efficient linear solvers. By establishing equivalence of error polynomials, we provide a route whereby acceleration techniques from numerical linear algebra may be applied to Gibbs sampling from normal distributions. The fastest solvers employ non- stationary iterations, hence the equivalent generalized Gibbs sampler induces an inhomogeneous Markov chain. Explicit calculation of the error polynomial guarantees convergence, while control of the error polynomial gives optimal performance. The adoption of the matrix splitting formalism gives the following practical benefits in the context of fixed-scan Gibbs sampling from normal targets: 1. a one-to-one equivalence between generalized Gibbs samplers and classical linear iterative solvers; 2. rates of convergence and error polynomials for the Markov chain induced by a generalized Gibbs sampler; 3. acceleration of the Gibbs sampler to induce an inhomogeneous Markov chain that achieves the optimal error polynomial, and hence has optimal convergence rate for expectations and in distribution; 4. numerical estimates of convergence rate of the (accelerated) Gibbs sampler in a single chain and a priori estimates of number of iterations to convergence; 5. access to preconditioning, whereby the sampling problem is transformed into an equivalent problem for which the accelerated Gibbs sampler has improved convergence rate. Some direct linear solvers have already been adapted to sampling from multivariate normal distributions, with Rue [52] demonstrating the use of solvers based on Cholesky factorization to allow computationally efficient sampling. This paper extends the connection to the iterative linear solvers. Since iterative methods are the most efficient for massive linear systems, the associated samplers will be the most efficient for very high dimensional normal targets. Polynomial accelerated Gibbs sampling 3713 1.1. Context and overview of results The Cholesky factorization is the conventional way to produce samples from a moderately sized multivariate normal distribution (Rue [52], Rue and Held [53]), and is also the preferred method for solving moderately sized linear systems. For large linear systems, iterative solvers are the methods of choice due to their inexpensive cost per iteration, and small computer memory re- quirements. Gibbs samplers applied to normal distributions are essentially identical to stationary itera- tive methods from numerical linear algebra. This connection was exploited by Adler [1], and independently by Barone and Frigessi [8], who noted that the component-wise Gibbs sampler is a stochastic version of the Gauss–Seidel linear solver, and accelerated the Gibbs sampler by introducing a relaxation parameter to implement the stochastic version of the successive over-relaxation (SOR) of Gauss–Seidel. This pairing was further analyzed by Goodman and Sokal [30]. This equivalence is depicted in panels A and B of Figure 1. Panel B shows the contours of a normal density π(x), and a sequence of coordinate-wise conditional samples taken by the Gibbs sampler applied to π. Panel A shows the contours of the quadratic minus log(π(x)) and the Gauss–Seidel sequence of coordinate optimizations,1 or, equivalently, solves of the normal equations ∇ log π(x) = 0. Note how in Gauss–Seidel the step sizes decrease towards conver- gence, which is a tell-tale sign that convergence (in value) is geometric. In Section 4, we will show that the iteration operator is identical to that of the Gibbs sampler in panel B, and hence the Gibbs sampler also converges geometrically (in distribution). Slow convergence of these al- gorithms is usually understood in terms of the same intuition; high correlations correspond to long narrow contours, and lead to small steps in coordinate directions and many iterations being required to move appreciably along the long axis of the target function. Roberts and Sahu [51] considered forward then backward sweeps of coordinate-wise Gibbs sampling, with relaxation parameter, to give a sampler they termed the REGS sampler. This is a stochastic version of the symmetric-SOR (SSOR) iteration, which comprises forward then backward sweeps of SOR. The equality of iteration operators and error polynomials, for these pairs of fixed-scan Gibbs samplers and iterative solvers, allows existing convergence results in numerical analysis texts (for example, Axelsson [5], Golub and Van Loan [29], Nevanlinna [45], Saad [54], Young [64]) to be used to establish convergence results for the corresponding Gibbs sampler. Existing results for rates of distributional convergence by fixed-sweep Gibbs samplers (Adler [1], Barone and Frigessi [8], Liu et al. [39], Roberts and Sahu [51]) may be established this way. The methods of Gauss–Seidel, SOR, and SSOR, give stationary linear iterations that were used as linear solvers in the 1950s, and are now considered very slow. The corresponding fixed-scan Gibbs
Load more

Recommended publications
  • Overview of Iterative Linear System Solver Packages
    Overview of Iterative Linear System Solver Packages Victor Eijkhout July, 1998 Abstract Description and comparison of several packages for the iterative solu- tion of linear systems of equations. 1 1 Intro duction There are several freely available packages for the iterative solution of linear systems of equations, typically derived from partial di erential equation prob- lems. In this rep ort I will give a brief description of a numberofpackages, and giveaninventory of their features and de ning characteristics. The most imp ortant features of the packages are which iterative metho ds and preconditioners supply; the most relevant de ning characteristics are the interface they present to the user's data structures, and their implementation language. 2 2 Discussion Iterative metho ds are sub ject to several design decisions that a ect ease of use of the software and the resulting p erformance. In this section I will give a global discussion of the issues involved, and how certain p oints are addressed in the packages under review. 2.1 Preconditioners A go o d preconditioner is necessary for the convergence of iterative metho ds as the problem to b e solved b ecomes more dicult. Go o d preconditioners are hard to design, and this esp ecially holds true in the case of parallel pro cessing. Here is a short inventory of the various kinds of preconditioners found in the packages reviewed. 2.1.1 Ab out incomplete factorisation preconditioners Incomplete factorisations are among the most successful preconditioners devel- op ed for single-pro cessor computers. Unfortunately, since they are implicit in nature, they cannot immediately b e used on parallel architectures.
    [Show full text]
  • Comparison of Two Stationary Iterative Methods
    Comparison of Two Stationary Iterative Methods Oleksandra Osadcha Faculty of Applied Mathematics Silesian University of Technology Gliwice, Poland Email:[email protected] Abstract—This paper illustrates comparison of two stationary Some convergence result for the block Gauss-Seidel method iteration methods for solving systems of linear equations. The for problems where the feasible set is the Cartesian product of aim of the research is to analyze, which method is faster solve m closed convex sets, under the assumption that the sequence this equations and how many iteration required each method for solving. This paper present some ways to deal with this problem. generated by the method has limit points presented in.[5]. In [9] presented improving the modiefied Gauss-Seidel method Index Terms—comparison, analisys, stationary methods, for Z-matrices. In [11] presented the convergence Gauss-Seidel Jacobi method, Gauss-Seidel method iterative method, because matrix of linear system of equations is positive definite, but it does not afford a conclusion on the convergence of the Jacobi method. However, it also proved I. INTRODUCTION that Jacobi method also converges. Also these methods were present in [12], where described each The term ”iterative method” refers to a wide range of method. techniques that use successive approximations to obtain more This publication present comparison of Jacobi and Gauss- accurate solutions to a linear system at each step. In this publi- Seidel methods. These methods are used for solving systems cation we will cover two types of iterative methods. Stationary of linear equations. We analyze, how many iterations required methods are older, simpler to understand and implement, but each method and which method is faster.
    [Show full text]
  • Chebyshev and Fourier Spectral Methods 2000
    Chebyshev and Fourier Spectral Methods Second Edition John P. Boyd University of Michigan Ann Arbor, Michigan 48109-2143 email: [email protected] http://www-personal.engin.umich.edu/jpboyd/ 2000 DOVER Publications, Inc. 31 East 2nd Street Mineola, New York 11501 1 Dedication To Marilyn, Ian, and Emma “A computation is a temptation that should be resisted as long as possible.” — J. P. Boyd, paraphrasing T. S. Eliot i Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series .................................... 20 2.3 Orders of Convergence ............................... 25 2.4 Convergence Order ................................. 27 2.5 Assumption of Equal Errors ...........................
    [Show full text]
  • Iterative Solution Methods and Their Rate of Convergence
    Uppsala University Graduate School in Mathematics and Computing Institute for Information Technology Numerical Linear Algebra FMB and MN1 Fall 2007 Mandatory Assignment 3a: Iterative solution methods and their rate of convergence Exercise 1 (Implementation tasks) Implement in Matlab the following iterative schemes: M1: the (pointwise) Jacobi method; M2: the second order Chebyshev iteration, described in Appendix A; M3: the unpreconditioned conjugate gradient (CG) method (see the lecture notes); M4: the Jacobi (diagonally) preconditioned CG method (see the lecture notes). For methods M3 and M4, you have to write your own Matlab code. For the numerical exper- iments you should use your implementation and not the available Matlab function pcg. The latter can be used to check the correctness of your implementation. Exercise 2 (Generation of test data) Generate four test matrices which have different properties: A: A=matgen disco(s,1); B: B=matgen disco(s,0.001); C: C=matgen anisot(s,1,1); D: D=matgen anisot(s,1,0.001); Here s determines the size of the problem, namely, the matrix size is n = 2s. Check the sparsity of the above matrices (spy). Compute the complete spectra of A, B, C, D (say, for s = 3) by using the Matlab function eig. Plot those and compare. Compute the condition numbers of these matrices. You are advised to repeat the experiment for a larger s to see how the condition number grows with the matrix size n. Exercise 3 (Numerical experiments and method comparisons) Test the four meth- ods M1, M2, M3, M4 on the matrices A, B, C, D.
    [Show full text]
  • Iterative Methods for Linear Systems of Equations: a Brief Historical Journey
    Iterative methods for linear systems of equations: A brief historical journey Yousef Saad Abstract. This paper presents a brief historical survey of iterative methods for solving linear systems of equations. The journey begins with Gauss who developed the first known method that can be termed iterative. The early 20th century saw good progress of these methods which were initially used to solve least-squares systems, and then linear systems arising from the discretization of partial differential equations. Then iterative methods received a big impetus in the 1950s - partly because of the development of computers. The survey does not attempt to be exhaustive. Rather, the aim is to bring out the way of thinking at specific periods of time and to highlight the major ideas that steered the field. 1. It all started with Gauss A big part of the contributions of Carl Friedrich Gauss can be found in the vo- luminous exchanges he had with contemporary scientists. This correspondence has been preserved in a number of books, e.g., in the twelve \Werke" volumes gathered from 1863 to 1929 at the University of G¨ottingen1. There are also books special- ized on specific correspondences. One of these is dedicated to the exchanges he had with Christian Ludwig Gerling [67]. Gerling was a student of Gauss under whose supervision he earned a PhD from the university of G¨ottingenin 1812. Gerling later became professor of mathematics, physics, and astronomy at the University of Marburg where he spent the rest of his life from 1817 and maintained a relation with Gauss until Gauss's death in 1855.
    [Show full text]
  • Linear Algebra Libraries
    Linear Algebra Libraries Claire Mouton [email protected] March 2009 Contents I Requirements 3 II CPPLapack 4 III Eigen 5 IV Flens 7 V Gmm++ 9 VI GNU Scientific Library (GSL) 11 VII IT++ 12 VIII Lapack++ 13 arXiv:1103.3020v1 [cs.MS] 15 Mar 2011 IX Matrix Template Library (MTL) 14 X PETSc 16 XI Seldon 17 XII SparseLib++ 18 XIII Template Numerical Toolkit (TNT) 19 XIV Trilinos 20 1 XV uBlas 22 XVI Other Libraries 23 XVII Links and Benchmarks 25 1 Links 25 2 Benchmarks 25 2.1 Benchmarks for Linear Algebra Libraries . ....... 25 2.2 BenchmarksincludingSeldon . 26 2.2.1 BenchmarksforDenseMatrix. 26 2.2.2 BenchmarksforSparseMatrix . 29 XVIII Appendix 30 3 Flens Overloaded Operator Performance Compared to Seldon 30 4 Flens, Seldon and Trilinos Content Comparisons 32 4.1 Available Matrix Types from Blas (Flens and Seldon) . ........ 32 4.2 Available Interfaces to Blas and Lapack Routines (Flens and Seldon) . 33 4.3 Available Interfaces to Blas and Lapack Routines (Trilinos) ......... 40 5 Flens and Seldon Synoptic Comparison 41 2 Part I Requirements This document has been written to help in the choice of a linear algebra library to be included in Verdandi, a scientific library for data assimilation. The main requirements are 1. Portability: Verdandi should compile on BSD systems, Linux, MacOS, Unix and Windows. Beyond the portability itself, this often ensures that most compilers will accept Verdandi. An obvious consequence is that all dependencies of Verdandi must be portable, especially the linear algebra library. 2. High-level interface: the dependencies should be compatible with the building of the high-level interface (e.
    [Show full text]
  • Implementing a Preconditioned Iterative Linear Solver Using Massively Parallel Graphics Processing Units
    Implementing a Preconditioned Iterative Linear Solver Using Massively Parallel Graphics Processing Units by Amirhassan Asgari Kamiabad A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2011 by Amirhassan Asgari Kamiabad Abstract Implementing a Preconditioned Iterative Linear Solver Using Massively Parallel Graphics Processing Units Amirhassan Asgari Kamiabad Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2011 The research conducted in this thesis provides a robust implementation of a precondi- tioned iterative linear solver on programmable graphic processing units (GPUs), appro- priately designed to be used in electric power system analysis. Solving a large, sparse linear system is the most computationally demanding part of many widely used power system analysis, such as power flow and transient stability. This thesis presents a de- tailed study of iterative linear solvers with a focus on Krylov-based methods. Since the ill-conditioned nature of power system matrices typically requires substantial precon- ditioning to ensure robustness of Krylov-based methods, a polynomial preconditioning technique is also studied in this thesis. Programmable GPUs currently offer the best ratio of floating point computational throughput to price for commodity processors, outdistancing same-generation CPUs by an order of magnitude. This has led to the widespread
    [Show full text]
  • 11 Mar 2021 Acceleration Methods
    Acceleration Methods Alexandre d’Aspremont CNRS & Ecole Normale Supérieure, Paris [email protected] Damien Scieur Samsung SAIT AI Lab & MILA, Montreal [email protected] Adrien Taylor INRIA & Ecole Normale Supérieure, Paris [email protected] arXiv:2101.09545v2 [math.OC] 11 Mar 2021 Contents 1 Introduction 2 2 Chebyshev Acceleration 5 2.1 Introduction ................................... 5 2.2 OptimalMethodsandMinimaxPolynomials . ..... 7 2.3 TheChebyshevMethod ............................. 9 3 Nonlinear Acceleration 15 3.1 Introduction ................................... 15 3.2 QuadraticCase ................................. 16 3.3 The Non-Quadratic Case: Regularized Nonlinear Acceleration.. .. .. 20 3.4 Constrained/Proximal Nonlinear Acceleration . .......... 26 3.5 Heuristics for Speeding-up Nonlinear Acceleration . ............ 29 3.6 RelatedWork .................................. 31 4 Nesterov Acceleration 32 4.1 Introduction ................................... 33 4.2 GradientMethodandPotential Functions . ...... 35 4.3 OptimizedGradientMethod . 38 4.4 Nesterov’sAcceleration . 45 4.5 AccelerationunderStrongConvexity . ...... 51 4.6 Recent Variants of Accelerated Methods . ...... 59 4.7 PracticalExtensions . .. .. .. .. .. .. .. .. 65 4.8 Notes&References ............................... 81 5 Proximal Acceleration and Catalyst 85 5.1 Introduction ................................... 85 5.2 Proximal Point Algorithm and Acceleration . ........ 86 5.3 Güler and Monteiro-Svaiter Acceleration . ........ 90 5.4 ExploitingStrongConvexity.
    [Show full text]
  • Explicit Iterative Methods of Second Order and Approximate Inverse Preconditioners for Solving Complex Computational Problems
    Applied Mathematics, 2020, 11, 307-327 https://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385 Explicit Iterative Methods of Second Order and Approximate Inverse Preconditioners for Solving Complex Computational Problems Anastasia-Dimitra Lipitakis Department of Informatics and Telematics, Harokopio University, Athens, Greece How to cite this paper: Lipitakis, A.-D. Abstract (2020) Explicit Iterative Methods of Second Order and Approximate Inverse Precondi- Explicit Exact and Approximate Inverse Preconditioners for solving complex tioners for Solving Complex Computational linear systems are introduced. A class of general iterative methods of second Problems. Applied Mathematics, 11, 307-327. order is presented and the selection of iterative parameters is discussed. The https://doi.org/10.4236/am.2020.114023 second order iterative methods behave quite similar to first order methods Received: March 3, 2020 and the development of efficient preconditioners for solving the original li- Accepted: April 19, 2020 near system is a decisive factor for making the second order iterative methods Published: April 22, 2020 superior to the first order iterative methods. Adaptive preconditioned Con- jugate Gradient methods using explicit approximate preconditioners for Copyright © 2020 by author(s) and Scientific Research Publishing Inc. solving efficiently large sparse systems of algebraic equations are also pre- This work is licensed under the Creative sented. The generalized Approximate Inverse Matrix techniques can be effi- Commons Attribution International ciently used in conjunction with explicit iterative schemes leading to effective License (CC BY 4.0). composite semi-direct solution methods for solving large linear systems of http://creativecommons.org/licenses/by/4.0/ algebraic equations.
    [Show full text]
  • Solution of General Linear Systems of Equations Using Block Krylov Based
    Solution of general linear systems of equations using blo ck Krylov based iterative metho ds on distributed computing environments LeroyAnthony Drummond Lewis Decemb er i Resolution de systemes lineaires creux par des metho des iteratives par blo cs dans des environnements distribues heterogenes Resume Nous etudions limplantation de metho des iteratives par blo cs dans des environnements mul tipro cesseur amemoire distribuee p our la resolution de systemes lineaires quelconques Dans un premier temps nous nous interessons aletude du p otentiel de la metho de du gradient con jugue classique en environnement parallele Dans un deuxieme temps nous etudions une autre metho de iterativedelameme famille que celle du gradient conjugue qui a ete concue p our resoudre simultanementdessystemes lineaires avec de multiples seconds membres et commune mentreferencee sous le nom de gradient conjugue par blo cs La complexite algorithmique de la metho de du gradient conjugue par blo cs est superieure a celle de la metho de du gradientconjugue classique car elle demande plus de calculs par iteration et necessite en outre plus de memoire Malgre cela la metho de du gradient conjugue par blo cs apparat comme etant mieux adaptee aux environnements vectoriels et paralleles Nous avons etudie trois variantes du gradient conjugue par blo cs basees sur dierents mo deles de programmation parallele et leur ecaciteaete comparee sur divers environnements distribues Pour la resolution des systemes lineaires non symetriques nous considerons lutilisation de me thodes
    [Show full text]
  • LSRN: a Parallel Iterative Solver for Strongly Over-Or Under-Determined
    LSRN: A PARALLEL ITERATIVE SOLVER FOR STRONGLY OVER- OR UNDER-DETERMINED SYSTEMS XIANGRUI MENG∗, MICHAEL A. SAUNDERSy , AND MICHAEL W. MAHONEYz Abstract. We describe a parallel iterative least squares solver named LSRN that is based on n random normal projection. LSRN computes the min-length solution to minx2R kAx − bk2, where m×n A 2 R with m n or m n, and where A may be rank-deficient. Tikhonov regularization may also be included. Since A is only involved in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and LSRN automatically speeds up when A is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size dγ min(m; n)e × min(m; n), where γ is moderately larger than 1, e.g., γ = 2. We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large problems on clusters with high communication cost. Numerical results demonstrate that on a shared-memory machine, LSRN outperforms LAPACK's DGELSD on large dense problems, and MATLAB's backslash (SuiteSparseQR) on sparse problems. Further experiments demonstrate that LSRN scales well on an Amazon Elastic Compute Cloud cluster. Key words. linear least squares, over-determined system, under-determined system, rank- deficient, minimum-length solution, LAPACK, sparse matrix, iterative method, preconditioning, LSQR, Chebyshev semi-iterative method, Tikhonov regularization, ridge regression, parallel com- puting, random projection, random sampling, random matrix, randomized algorithm AMS subject classifications.
    [Show full text]
  • A Parallel Iterative Solver for Strongly Over- Or Underdetermined Systems∗
    SIAM J. SCI. COMPUT. c 2014 Society for Industrial and Applied Mathematics Vol. 36, No. 2, pp. C95–C118 LSRN: A PARALLEL ITERATIVE SOLVER FOR STRONGLY OVER- OR UNDERDETERMINED SYSTEMS∗ † ‡ § XIANGRUI MENG , MICHAEL A. SAUNDERS , AND MICHAEL W. MAHONEY Abstract. We describe a parallel iterative least squares solver named LSRN that is based on random normal projection. LSRN computes the min-length solution to minx∈Rn Ax − b2,where A ∈ Rm×n with m n or m n,andwhereA may be rank-deficient. Tikhonov regularization may also be included. Since A is involved only in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and LSRN automatically speeds up when A is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size γ min(m, n)×min(m, n), where γ is moderately larger than 1, e.g., γ = 2. We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large prob- lems on clusters with high communication cost. Numerical results show that on a shared-memory machine, LSRN is very competitive with LAPACK’s DGELSD and a fast randomized least squares solver called Blendenpik on large dense problems, and it outperforms the least squares solver from SuiteSparseQR on sparse problems without sparsity patterns that can be exploited to reduce fill-in.
    [Show full text]